An Unusual Approach to Probability

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Math Lab 21

• An Unusual Approach to Probability

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Purpose of Lab

• Explore the idea that the sum of all
  probabilities is one, and then use it to solve a
  problem.

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Math Principles

• Probability, sum of all probabilities,
  theoretical probability, experimental
  probability, solving a linear equation.

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What Its All About

• We are interested in when the spinners stop on
  the red sections. The probability of getting a
  red with the four-segment spinner is 1/4 since
  there are four different colored segments but
  only one that is red. Similarly, the five-segment
  spinners probability of coming up red is 1/5.

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What Its All About

• The problem we are to solve here is to find the
  probability of getting at least one red when
  considering both spinners simultaneously.

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An Unusual Approach to Probability

• Spin both devices and record a tally on the board
  of both the total number of trials and the number
  of successes. Success could either be just one
  red or both red.

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An Unusual Approach to Probability

• Spin both devices and record a tally on the board
  of both the total number of trials and the number
  of successes. Success could either be just one
  red or both red.
• Trials Successes

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An Unusual Approach to Probability

• Theoretical determination The very heart of this
  lab is the fact that the sum of the probabilities
  of all possibilities is one.
• As an example, consider a weather prediction of
  75 rain. Of course, this means a 25 chance of
  no rain. These two probabilities add to give 100
  which is equivalent to a probability of one.

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An Unusual Approach to Probability

• With regard to getting a red with the two
  spinners, all possible probabilities are
•  
• P0 probability of no reds
• P1 probability of exactly one red
• P2 probability of two reds

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An Unusual Approach to Probability

• With regard to getting a red with the two
  spinners, all possible probabilities are
•  
• P0 probability of no reds
• P1 probability of exactly one red
• P2 probability of two reds
• Expressing the sum of these probabilities, we get
  the fundamental equation with which we will solve
  our problem
• P0 P1 P2 1

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An Unusual Approach to Probability

• At this point it is important to realize that the
  probability we want (the probability of getting
  at least one red) is the probability of getting
  exactly one red plus the probability of getting
  two reds. This can be expressed as
• P1 P2

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An Unusual Approach to Probability

• Rewrite the sum equation, isolate these two
  terms in parenthesis, and then subtract P0 from
  both sides

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An Unusual Approach to Probability

• Rewrite the sum equation, isolate these two
  terms in parenthesis, and then subtract P0 from
  both sides
•  
• P0 (P1 P2)1

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An Unusual Approach to Probability

• Rewrite the sum equation, isolate these two
  terms in parenthesis, and then subtract P0 from
  both sides
•  
• P0 (P1 P2)1
• (P1 P2)1 P0

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An Unusual Approach to Probability

• Instead of finding two probabilities (P1 P2)
  and adding them, we have the easier task of
  finding P0 and subtracting it from 1
•  
• P0 (prob of no red on spinner5) X (prob of no
  red on spinner4)
• P0
• P0

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An Unusual Approach to Probability

• Finally, the solution is
• P1 P2